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- TL;DR Summary
- Here, I present a few silly doubts on how to define the maximum number of solutions of a polynomial using set notation and theory.
Let ##Q_{n}(x)## be the inverse of an nth-degree polynomial. Precisely,
$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,
It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing how to proceed, below you may find my attempt.
Let the maximum number of solutions of ##Q^{-1}_{n}(x)=0## be
$$J_{n}=\text{Sup}\{\pi(Q^{-1}_{n}(x)=0):\partial Q_{n}^{-1}(x) \leq n\}$$,
in which ##\partial## denotes "the degree of" and ##\pi(Q^{-1}_{n}(x)=0)## is the number of solutions of ##Q^{-1}_{n}(x)=0##.
Based on the above, I ask:
1. Is the above definition correct?
2. How may improve and formally define ##J_{n}## using proper notation of set theory and mathematics? That is to say, is there anything else that I should consider to define ##J_{n}## in the way stated above?
3. In order to define the degree of a polynomial, should I consider ##n \in \mathbb{N}## or ##n \in \mathbb{Z}##?
Thanks in advance.
$$Q_{n}(x)=\displaystyle\frac{1}{P_{n}(x)}$$,
It is of my interest to use the set notation to formally define a number, ##J_{n}## that provides the maximum number of solutions of ##Q_{n}(x)^{-1}=0##. Despite not knowing how to proceed, below you may find my attempt.
Let the maximum number of solutions of ##Q^{-1}_{n}(x)=0## be
$$J_{n}=\text{Sup}\{\pi(Q^{-1}_{n}(x)=0):\partial Q_{n}^{-1}(x) \leq n\}$$,
in which ##\partial## denotes "the degree of" and ##\pi(Q^{-1}_{n}(x)=0)## is the number of solutions of ##Q^{-1}_{n}(x)=0##.
Based on the above, I ask:
1. Is the above definition correct?
2. How may improve and formally define ##J_{n}## using proper notation of set theory and mathematics? That is to say, is there anything else that I should consider to define ##J_{n}## in the way stated above?
3. In order to define the degree of a polynomial, should I consider ##n \in \mathbb{N}## or ##n \in \mathbb{Z}##?
Thanks in advance.